Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: A smooth picture proves nothing — page 25

Page 25 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 25 of 31.

The fuselage's overspeed reaches along the wing. The extra axial speed the fuselage alone gives the air beside it, as a fraction of the flight speed, against distance out along the wing from the fuselage's axis in fuselage radii, level with its thickest section, at Mach 0, 0.6 and 0.8. At the fuselage's side it is 2.1 per cent at low speed and 2.5 at Mach 0.8; it falls only to two-thirds of that two radii out and a quarter at five — a long body's middle " + "spreads like a line of sources, not a point — and the section beside it is already running at twelve per cent over at low speed and twenty over at Mach 0.8. Ideal flow

A fuselage lowers its wing root's critical Mach number

A slender fuselage barely disturbs the air: two per cent of overspeed at its side, against the twelve a wing section of ordinary thickness makes. At the wing root the two are added, and two per cent on top of twelve moves the root's critical Mach number down by more than a hundredth — as much as making the whole wing thirteen per cent thicker. The fuselage's overspeed also fades far more slowly along the span than its size suggests, because the middle of a long body spreads its disturbance like a line, not a point.

 A quiet flame has one cusp; a noisy one keeps making more. Two flame fronts twenty neutral wavelengths wide, burning upwards, drawn apart for clarity: above, the quiet front, the exact pole solution with one cusp per period and the most poles it can hold; below, the same flame with a noise of a millionth kicking its growing wavelengths. The noise keeps seeding small wrinkles on the smooth arcs; each grows as it is swept along the arc into the cusp, so the noisy front carries a train of sub-cusps the quiet one never has, and advances at 0.862 against the quiet 0.5. Flows and fields

A flame's speed limit holds only in silence

A quiet wrinkled flame settles into a single cusp and a speed it never exceeds, however wide it grows. Add noise and the limit is gone. A disturbance of one part in a million million already speeds the front up; one in a million makes it a third faster; and a noisy flame, unlike a quiet one, keeps getting faster as it gets wider, because every added width is room for more wrinkles to be born on its arcs and swept into its cusp. The quiet flame's speed is a property of a silence no burner has.

 Below the capillary length, the coat is set by the fibre, not the bath. The coat's thickness in units of ℓ_c Ca^(2/3) against the fibre's radius in capillary lengths, on logarithmic axes: the dynamic meniscus's 1.3376 over the curvature the static meniscus asks of it, 1/b + Z/ℓ_c². Thin fibres take Quéré's 1.3376 b Ca^(2/3), set by their own radius; thick ones take Landau–Levich's 0.9458 ℓ_c Ca^(2/3). The two laws cross at b = 0.71 ℓ_c, where the coat is 0.61 of either; at b = 0.1 ℓ_c it is 0.137 of what the plate's law would give. Viscosity

A thin fibre coats by its own radius, and beads by it

A plate drawn out of a bath carries a film set by the capillary length, the size at which surface tension and gravity balance. A fibre thinner than that length carries a film set by its own radius instead, often a tenth of what the plate's law promises — and the same radius then decides how quickly that film gathers into beads. The faster the fibre is drawn, the thicker its coat and the shorter the length of it that comes out smooth.

 Gas takes the top off a tall, brief pulse. The head at a closed valve against time, for a line whose margin to vapour pressure is 0.49 Joukowsky rises: the exact vapour-cavity history, whose first pulse after the collapse reaches 1.94 rises for only 0.028 of a round trip, and the same line with a pocket of free gas at the valve of a millionth, a hundred-thousandth and a ten-thousandth of the pipe's volume. A hundred-thousandth brings the pulse down to 1.62: the pocket cannot be squeezed fast enough to follow a pulse that short. Fluids at work

Air at a valve softens the hammer only in quantity

A vapour cavity at a closed valve holds the head at one value, and that is why the pressure after it collapses climbs a staircase of equal steps. Put a pocket of free air there instead and the valve becomes a spring. A millionth of the pipe's volume trims only the tallest, briefest pulses; a hundred-thousandth takes a third off them and moves them; a ten-thousandth can make the pulse taller than it was, even on a line that never cavitates. The air that reliably removes the hammer is a thousandth of the pipe, which is a deliberate air vessel rather than a little dissolved gas.

 Past its threshold a ridge slides at a speed its angles set. The steady speed of a ridge of liquid one square capillary length in cross-section, as a capillary number, against the plate's tilt, on three surfaces. Below each surface's threshold it is stuck. Past it the speed rises from zero, linearly at first, as far as the tilt allows. On clean glass the curve barely exists: between its threshold and the speed at which its uphill contact line fails there is a sliver of tilt, and a ridge pushed past that cannot slide steadily with a clean trailing edge. Regimes and numbers

A sliding drop is held harder the faster it goes

A ridge of liquid on a tilted plate starts to slide when its weight beats the difference between its two contact angles' cosines. Once it moves, the angles move too: the front steepens and the back flattens, by a law set in the viscous corners at each edge. So the resistance rises with speed from exactly the static value, and a sliding drop has no kinetic friction lower than its static one — it stops at the tilt it started at. The back edge's angle falls to nothing at a finite speed, and past that no drop slides with a clean back.

 Bursts in the velocity put the tails back. The kurtosis of the pairs' separation against how long each pair remembers its velocity's direction, β, in turnover times. The lowest curve is a Gaussian velocity, as in the earlier calculation. The others give the velocity a flatness of 4, as measured in the inertial range, with its amplitude remembered for α turnover times. At the memory real pairs are estimated to have, β = 0.7, the Gaussian cloud's kurtosis is 1.89; with bursts remembered for three turnover times it is 3.41, and with the amplitude frozen 4.4 — either side of Richardson's 3.76, dashed. Transition and turbulence

Bursts and memory pull a cloud both ways

A pair of fluid particles that remembers its relative velocity spreads into a cloud with shorter tails than Richardson's, and the earlier calculation proposed reading the memory off the cloud's shape. Real relative velocities come in bursts, their amplitude set by a local dissipation that varies, and a burst that lasts pushes the tails back out — hard, because separation grows as the cube of diffusivity. At the memory real pairs are estimated to have, the two effects nearly cancel, and a cloud can have Richardson's exact shape for entirely the wrong reason.

 An S-shaped set of states and a slow variable that crosses it. The film's steady mean temperature against the housing's, on the cool and hot branches of operating points, with the line on which the housing is in equilibrium with the film, at a derating of 0.22 per unit. Between housing temperatures 1.24 and 1.59 both branches exist. At the derating the essay uses, 0.22, neither meets the equilibrium line there: on the hot branch the housing is always warming, on the cool branch always cooling, so the film runs round the loop drawn over them, jumping at each fold. Viscosity

A slow sensor makes the bearing cycle

Derate the motor driving a self-heating oil film on the temperature of its housing, which warms over minutes, and a protection meant to hold the bearing cool instead makes it cycle for ever: jump hot, warm the housing until the hot state is lost, drop cool, let the housing cool until the cool state is lost, jump again. The period belongs to the housing, the amplitude belongs to the film, and the derating strength decides only whether the cycle happens at all.

 The Earth moves the whole train to the right. The centreline of a river like the lower Ob, in its own widths, flowing left to right with its right bank below: now, and 100 years later with and without the Earth's rotation. In that time the bends grow and move 0.77 widths downstream, identically in both. The Earth adds only a steady shift towards the right bank, 0.39 widths a century — the bend migration rate times the Earth's 39% share of the helix. What is taught wrongly

A river drifts right, and only a reach can show it

Add the Earth's rotation to the law by which a meandering river migrates and it changes nothing about the meanders: they grow and travel downstream exactly as before. It adds one thing, a steady sideways drift of the whole river towards its right bank, at the bend migration rate times the Earth's share of the helix — four-tenths of a width a century on the lower Ob. No single bend can show it, because each bend moves several times further on its own. A survey of about ten to twenty bends can.

 A flow's window mean converges a power faster. The variance of the mean over an L × L window, as a fraction of the point variance, against the window's side in integral lengths, on logarithmic axes. The scalar field's falls as the inverse area. The plane cut through a three-dimensional flow falls the same way at half the level. The velocity of a two-dimensional incompressible flow falls as the inverse cube of the side, because its mean over any window is a streamfunction difference around the window's edge. Lines are closed forms; dots are Monte Carlo means over fifty generated fields. Transition and turbulence

A snapshot counts areas, and a flow counts its edge

A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.

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